On Horowitz and Shelah's Borel maximal eventually different family

TL;DR

This paper demonstrates the existence of a closed, effectively closed, and Borel maximal eventually different family within ZF set theory, advancing understanding of such families' definability and complexity.

Contribution

It constructs a closed and effectively closed (Pi^0_1) maximal eventually different family in ZF, showing such families can be highly definable.

Findings

Existence of a closed, effectively closed, and Borel maximal eventually different family.

Construction works within ZF set theory without additional axioms.

Advances understanding of the complexity and definability of maximal eventually different families.

Abstract

We show there is a closed (in fact effectively closed, i.e., $\Pi^0_1$) eventually different family (working in ZF or less).